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---
:name: dsbgvx
:md5sum: 1ee65cfeddf2efe1e17ddc18f8adf220
:category: :subroutine
:arguments:
- jobz:
:type: char
:intent: input
- range:
:type: char
:intent: input
- uplo:
:type: char
:intent: input
- n:
:type: integer
:intent: input
- ka:
:type: integer
:intent: input
- kb:
:type: integer
:intent: input
- ab:
:type: doublereal
:intent: input/output
:dims:
- ldab
- n
- ldab:
:type: integer
:intent: input
- bb:
:type: doublereal
:intent: input/output
:dims:
- ldbb
- n
- ldbb:
:type: integer
:intent: input
- q:
:type: doublereal
:intent: output
:dims:
- ldq
- n
- ldq:
:type: integer
:intent: input
- vl:
:type: doublereal
:intent: input
- vu:
:type: doublereal
:intent: input
- il:
:type: integer
:intent: input
- iu:
:type: integer
:intent: input
- abstol:
:type: doublereal
:intent: input
- m:
:type: integer
:intent: output
- w:
:type: doublereal
:intent: output
:dims:
- n
- z:
:type: doublereal
:intent: output
:dims:
- ldz
- n
- ldz:
:type: integer
:intent: input
- work:
:type: doublereal
:intent: output
:dims:
- 7*n
- iwork:
:type: integer
:intent: output
:dims:
- 5*n
- ifail:
:type: integer
:intent: output
:dims:
- m
- info:
:type: integer
:intent: output
:substitutions:
ldz: "lsame_(&jobz,\"V\") ? MAX(1,n) : 1"
m: "lsame_(&range,\"A\") ? n : lsame_(&range,\"I\") ? iu-il+1 : 0"
ldq: "1 ? jobz = 'n' : MAX(1,n) ? jobz = 'v' : 0"
:fortran_help: " SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* DSBGVX computes selected eigenvalues, and optionally, eigenvectors\n\
* of a real generalized symmetric-definite banded eigenproblem, of\n\
* the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric\n\
* and banded, and B is also positive definite. Eigenvalues and\n\
* eigenvectors can be selected by specifying either all eigenvalues,\n\
* a range of values or a range of indices for the desired eigenvalues.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* JOBZ (input) CHARACTER*1\n\
* = 'N': Compute eigenvalues only;\n\
* = 'V': Compute eigenvalues and eigenvectors.\n\
*\n\
* RANGE (input) CHARACTER*1\n\
* = 'A': all eigenvalues will be found.\n\
* = 'V': all eigenvalues in the half-open interval (VL,VU]\n\
* will be found.\n\
* = 'I': the IL-th through IU-th eigenvalues will be found.\n\
*\n\
* UPLO (input) CHARACTER*1\n\
* = 'U': Upper triangles of A and B are stored;\n\
* = 'L': Lower triangles of A and B are stored.\n\
*\n\
* N (input) INTEGER\n\
* The order of the matrices A and B. N >= 0.\n\
*\n\
* KA (input) INTEGER\n\
* The number of superdiagonals of the matrix A if UPLO = 'U',\n\
* or the number of subdiagonals if UPLO = 'L'. KA >= 0.\n\
*\n\
* KB (input) INTEGER\n\
* The number of superdiagonals of the matrix B if UPLO = 'U',\n\
* or the number of subdiagonals if UPLO = 'L'. KB >= 0.\n\
*\n\
* AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N)\n\
* On entry, the upper or lower triangle of the symmetric band\n\
* matrix A, stored in the first ka+1 rows of the array. The\n\
* j-th column of A is stored in the j-th column of the array AB\n\
* as follows:\n\
* if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;\n\
* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).\n\
*\n\
* On exit, the contents of AB are destroyed.\n\
*\n\
* LDAB (input) INTEGER\n\
* The leading dimension of the array AB. LDAB >= KA+1.\n\
*\n\
* BB (input/output) DOUBLE PRECISION array, dimension (LDBB, N)\n\
* On entry, the upper or lower triangle of the symmetric band\n\
* matrix B, stored in the first kb+1 rows of the array. The\n\
* j-th column of B is stored in the j-th column of the array BB\n\
* as follows:\n\
* if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;\n\
* if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb).\n\
*\n\
* On exit, the factor S from the split Cholesky factorization\n\
* B = S**T*S, as returned by DPBSTF.\n\
*\n\
* LDBB (input) INTEGER\n\
* The leading dimension of the array BB. LDBB >= KB+1.\n\
*\n\
* Q (output) DOUBLE PRECISION array, dimension (LDQ, N)\n\
* If JOBZ = 'V', the n-by-n matrix used in the reduction of\n\
* A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,\n\
* and consequently C to tridiagonal form.\n\
* If JOBZ = 'N', the array Q is not referenced.\n\
*\n\
* LDQ (input) INTEGER\n\
* The leading dimension of the array Q. If JOBZ = 'N',\n\
* LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).\n\
*\n\
* VL (input) DOUBLE PRECISION\n\
* VU (input) DOUBLE PRECISION\n\
* If RANGE='V', the lower and upper bounds of the interval to\n\
* be searched for eigenvalues. VL < VU.\n\
* Not referenced if RANGE = 'A' or 'I'.\n\
*\n\
* IL (input) INTEGER\n\
* IU (input) INTEGER\n\
* If RANGE='I', the indices (in ascending order) of the\n\
* smallest and largest eigenvalues to be returned.\n\
* 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.\n\
* Not referenced if RANGE = 'A' or 'V'.\n\
*\n\
* ABSTOL (input) DOUBLE PRECISION\n\
* The absolute error tolerance for the eigenvalues.\n\
* An approximate eigenvalue is accepted as converged\n\
* when it is determined to lie in an interval [a,b]\n\
* of width less than or equal to\n\
*\n\
* ABSTOL + EPS * max( |a|,|b| ) ,\n\
*\n\
* where EPS is the machine precision. If ABSTOL is less than\n\
* or equal to zero, then EPS*|T| will be used in its place,\n\
* where |T| is the 1-norm of the tridiagonal matrix obtained\n\
* by reducing A to tridiagonal form.\n\
*\n\
* Eigenvalues will be computed most accurately when ABSTOL is\n\
* set to twice the underflow threshold 2*DLAMCH('S'), not zero.\n\
* If this routine returns with INFO>0, indicating that some\n\
* eigenvectors did not converge, try setting ABSTOL to\n\
* 2*DLAMCH('S').\n\
*\n\
* M (output) INTEGER\n\
* The total number of eigenvalues found. 0 <= M <= N.\n\
* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.\n\
*\n\
* W (output) DOUBLE PRECISION array, dimension (N)\n\
* If INFO = 0, the eigenvalues in ascending order.\n\
*\n\
* Z (output) DOUBLE PRECISION array, dimension (LDZ, N)\n\
* If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of\n\
* eigenvectors, with the i-th column of Z holding the\n\
* eigenvector associated with W(i). The eigenvectors are\n\
* normalized so Z**T*B*Z = I.\n\
* If JOBZ = 'N', then Z is not referenced.\n\
*\n\
* LDZ (input) INTEGER\n\
* The leading dimension of the array Z. LDZ >= 1, and if\n\
* JOBZ = 'V', LDZ >= max(1,N).\n\
*\n\
* WORK (workspace/output) DOUBLE PRECISION array, dimension (7*N)\n\
*\n\
* IWORK (workspace/output) INTEGER array, dimension (5*N)\n\
*\n\
* IFAIL (output) INTEGER array, dimension (M)\n\
* If JOBZ = 'V', then if INFO = 0, the first M elements of\n\
* IFAIL are zero. If INFO > 0, then IFAIL contains the\n\
* indices of the eigenvalues that failed to converge.\n\
* If JOBZ = 'N', then IFAIL is not referenced.\n\
*\n\
* INFO (output) INTEGER\n\
* = 0 : successful exit\n\
* < 0 : if INFO = -i, the i-th argument had an illegal value\n\
* <= N: if INFO = i, then i eigenvectors failed to converge.\n\
* Their indices are stored in IFAIL.\n\
* > N : DPBSTF returned an error code; i.e.,\n\
* if INFO = N + i, for 1 <= i <= N, then the leading\n\
* minor of order i of B is not positive definite.\n\
* The factorization of B could not be completed and\n\
* no eigenvalues or eigenvectors were computed.\n\
*\n\n\
* Further Details\n\
* ===============\n\
*\n\
* Based on contributions by\n\
* Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA\n\
*\n\
* =====================================================================\n\
*\n"
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